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Abstract

Threshold cryptography deals with situations where the authority to initiate or
perform cryptographic operations is distributed amongst a group of individuals.
Usually in these situations a secret sharing scheme is used to distribute shares
of a highly sensitive secret, such as the private key of a bank, to the involved
individuals so that only when a sufficient number of them can reconstruct the
secret but smaller coalitions cannot. The secret sharing problem was introduced
independently by Blakley and Shamir in 1979. They proposed two different solutions.
Both secret sharing schemes (SSS) are examples of linear secret sharing.
Many extensions and solutions based on these secret sharing schemes have appeared
in the literature, most of them using Shamir SSS. In this thesis, we apply
these ideas to Blakley secret sharing scheme.
Many of the standard operations of single-user cryptography have counterparts
in threshold cryptography. Function sharing deals with the problem of
distribution of the computation of a function (such as decryption or signature)
among several parties. The necessary values for the computation are distributed
to the participants using a secret sharing scheme. Several function sharing
schemes have been proposed in the literature with most of them using Shamir
secret sharing as the underlying SSS. In this work, we investigate how function
sharing can be achieved using linear secret sharing schemes in general and give
solutions of threshold RSA signature, threshold Paillier decryption and threshold
DSS signature operations. The threshold RSA scheme we propose is a generalization
of Shoup’s Shamir-based scheme. It is similarly robust and provably secure
under the static adversary model.
In threshold cryptography the authorization of groups of people are decided simply according to their size. There are also general access structures in which
any group can be designed as authorized. Multipartite access structures constitute
an example of general access structures in which members of a subset are
equivalent to each other and can be interchanged. Multipartite access structures
can be used to represent any access structure since all access structures are multipartite.
To investigate secret sharing schemes using these access structures,
we used Mignotte and Asmuth-Bloom secret sharing schemes which are based
on the Chinese remainder theorem (CRT). The question we tried to asnwer was
whether one can find a Mignotte or Asmuth-Bloom sequence for an arbitrary
access structure. For this purpose, we adapted an algorithm that appeared in the
literature to generate these sequences. We also proposed a new SSS which solves
the mentioned problem by generating more than one sequence.